On the Classical Prime Radical Formula and Classical Prime of Semimodules
Keywords:
prime subsemimodule, classical prime subsemimodule, prime radical, classical prime radical, prime idealAbstract
Let $R$ be a commutative semiring and $M$ an  $R$ semimodule. A proper subsemimodule  $N$ of  $M$ is called a classical prime subsemimodule, if for any  $a,b\in R$ and  $m\in M, abm\in N$ implies that  $am\in N$ or $bm\in N$.  We will introduce and study the notion of prime bases for classical prime subsemimodules and utilize them to derive some formulas on the classical prime radical of subsemimodules of a semimodule. In particular, we study some basic properties of prime radical and classical prime radical of subsemimodule in  $M$. Moreover, we investigate relationships between classical prime radical and prime radical of subsemimodule in $M$.
References
Atani R. E., 2010. Prime subsemimodules of semimodules, International Journal of Algebra. 4(26): 1299-1306.
Atani S. E. and Darani A. Y., 2006. On quasi-primary submodules, Chiang Mai J. Sci. 33(3): 249-254.
Baziar M. and Behboodi M., 2009. Classical primary submodules and decomposition theory of modules, J. Algebra Appl. 8(3): 351-362.
Behboodi M., Jahani-nezhad R. and Naderi M. H., 2011. Classical quasi-primary submodules, Bulletin of the Iranian Mathematical Society. 37(4): 51-71.
Dubey M. K. and Sarohe P., 2013. On 2-absorbing semimodules, Quasigroups and Related Systems. 21: 175 -184.
Ebrahimi Atani R., 2010. Prime subsemimodules of semimodules, Int. J.of Algebra. 4(26): 1299-1306.
Ebrahimi Atani R. and Ebrahimi Atani S., 2010. On subsemimodules of semimodules, Buletinul Academiei De Stiinte. 2(63): 20 - 30.
Ebrahimi Atani S. and Shajari Kohan M., 2010. A note on finitely generated multiplication semimodules over comutative semirings, International Journal of Algebra. 4(8): 389-396.
Fuchs L., 1947. On quasi-primary ideals, Acta Univ. Szeged. Sect. Sci. Math. 11: 174-183.
Saffar Ardabili J., Motmaen S. and Yousefian Darani A., 2011. The spectrum of classical prime subsemimodules, Australian Journal of Basic and Applied Sciences. 5(11): 1824-1830.
Srinivasa Reddy M., Amarendra Babu V. and Srinivasa Rao P. V., 2013. Weakly primary subsemimodules of partial semimodules, International Journal of Mathematics and Computer Applications Research (IJMCAR), 3: 45- 56.
Tavallaee H.A. and Zolfaghari M., 2012. Some remarks on weakly prime and weakly semiprime submodules, Journal of Advanced Research in Pure Math., 4(1): 19 - 30.
Tavallaee H.A. and Zolfaghari M., 2013. On semiprime submodules and related results, J. Indones. Math. Soc., 19(1): 49-59.
Yesilot G., Oral, K. H. and Tekir, U., 2010. On prime subsemimodules of semimodules, International Journal of Algebra, 4(1): 53-60.
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